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The relative terms H ∗ (Z2 ; K0 (R)) in the Rothenberg exact sequences relating the free and projective L-groups of R are the same for the symmetric and quadratic L-groups . . −−→ Lnh (R) −−→ Lnp (R) −−→ H n (Z2 ; K0 (R)) −−→ Ln−1 (R) −−→ . . , h . . −−→ Lhn (R) −−→ Lpn (R) −−→ H n (Z2 ; K0 (R)) −−→ Lhn−1 (R) −−→ . . Thus the free and projective hyperquadratic L-groups of R coincide L∗ (R) = L∗h (R) = L∗p (R) . Similarly, the hyperquadratic L-groups of the categories Ah (R) and Ap (R) coincide, being the 4-periodic versions of the hyperquadratic L-groups L∗ (R) Ln (Ah (R)) = Ln (Ap (R)) = lim Ln+4k (R) (n ∈ Z) , −→ k the direct limits being taken with respect to the double skew-suspension maps.

G. projective (left) R-modules. ¯ r) , e(P )−1 : P −−→ P ∗∗ ; x −−→ (f −−→ f (x)) . g. y) . The duality isomorphism TP,Q : P ⊗Ap (R) Q−−→Q ⊗Ap (R) P corresponds to the transposition isomorphism TP,Q : P ⊗R Q −−→ Q ⊗R P ; x ⊗ y −−→ y ⊗ x . g. free R-modules. 4 Given a commutative ring R, a group π and a group morphism w: π−−→{±1} let R[π]w denote the group ring R[π] with the w-twisted involution ¯: R[π]w −−→ R[π]w ; a = rg g −−→ a ¯ = rg w(g)g −1 (rg ∈ R) . g∈π g∈π 1. Algebraic Poincare complexes 29 This is the example occurring most frequently in topological applications, with w an orientation character.

2 (i) A chain bundle (C, γ) is a chain complex C in A together with a 0-cycle γ ∈ (W % T C)0 . (ii) A map of chain bundles in A (f, b) : (C, γ) −−→ (C , γ ) is a chain map f : C−−→C together with a 1-chain b ∈ (W % T C)1 such that f % (γ ) − γ = d(b) ∈ (W % T C)0 . For any chain complex C in A there is defined a suspension isomorphism S : W % C −−→ S −1 (W % SC) ; θ −−→ Sθ sending an n-chain θ ∈ (W % C)n to the (n + 1)-chain Sθ ∈ (W % SC)n+1 with (Sθ)t = θt−1 : (SC)n−r+t+1 = C n−r+t −−→ (SC)r = Cr−1 .

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Algebraic L theory and topological manifolds by Ranicki

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